Examine the product of the two matrices to determine if each...

Examine the product of the two matrices to determine if each is the inverse of the other.
$$\left[\begin{array}{rrr}1 & 2 & 0 \\0 & 1 & 0 \\-1 & -1 & 1\end{array}\right] ;\left[\begin{array}{rrr}1 & -2 & 0 \\0 & 1 & 0 \\1 & -1 & 0\end{array}\right]$$

Key Concepts

Inverse Matrix

An inverse matrix of a given square matrix is another matrix such that when they are multiplied together, the result is the identity matrix. This concept is central in linear algebra and is used in solving linear systems, among other applications. Determining whether two matrices are inverses involves verifying that their product in both orders (AB and BA) equals the identity matrix.

Matrix Multiplication

Matrix multiplication is a binary operation where two matrices are combined to produce a new matrix. Each entry in the resulting matrix is obtained by taking the sum of products of corresponding entries from the rows of the first matrix and the columns of the second matrix. This operation is crucial in verifying matrix inverses because the product of a matrix and its inverse must yield the identity matrix according to the definition of inversion.

Identity Matrix

An identity matrix is a special square matrix that has ones on its main diagonal and zeros elsewhere. It serves as the multiplicative identity in matrix algebra, meaning that when any square matrix is multiplied by an appropriately sized identity matrix, the original matrix is unchanged. In the context of matrix inverses, if the product of two matrices results in the identity matrix, each matrix is the inverse of the other.

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